Math / Trigonometry

QuestionSelect the correct answer.
In right triangle $A B C, \angle A$ and $\angle \boldsymbol{B}$ are complementary angles and $\sin A=\frac{B}{9}$ . What is $\cos B$ ? A. $\frac{8 \sqrt{17}}{17}$ B. $\frac{8}{9}$ C. $\frac{\sqrt{17}}{9}$ D. $\frac{\sqrt{17}}{8}$

Studdy Solution

STEP 1

What is this asking? If the sine of angle A is 8/9, and angles A and B are complementary in a right triangle, what's the cosine of angle B? Watch out! Don't mix up sine and cosine, and remember what "complementary" means!

STEP 2

1. Define Complementary Angles
2. Relate Sine and Cosine
3. Calculate Cosine B

STEP 3

Alright, so we're dealing with complementary angles.
What does that even mean?
It means they add up to $\mathbf{90^\circ}$ !
Since we're in a right triangle, and one angle is $\mathbf{90^\circ}$ , the other two must be complementary.
So, we know $\angle A + \angle B = 90^\circ$ .

STEP 4

Now, here's the magical connection between sine and cosine of complementary angles: the sine of an angle is equal to the cosine of its complement!
Think of it this way: in a right triangle, if you switch your focus from one acute angle to the other, the opposite side becomes the adjacent side, and voilà!
Sine turns into cosine.

STEP 5

Mathematically, we can write this as $\sin(A) = \cos(90^\circ - A)$ .
Since $\angle A + \angle B = 90^\circ$ , we can rewrite this as $90^\circ - A = B$ .
Therefore, $\sin(A) = \cos(B)$ .

STEP 6

We're given that $\sin(A) = \frac{8}{9}$ .
And we just figured out that $\sin(A) = \cos(B)$ .
So, guess what?
We're practically done!

STEP 7

If $\sin(A) = \frac{8}{9}$ and $\sin(A) = \cos(B)$ , then $\cos(B)$ must also be $\frac{8}{9}$ !

STEP 8

So, the cosine of angle B is $\frac{8}{9}$ , which corresponds to answer choice B!

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QuestionSelect the correct answer.
In right triangle $A B C, \angle A$ and $\angle \boldsymbol{B}$ are complementary angles and $\sin A=\frac{B}{9}$ . What is $\cos B$ ? A. $\frac{8 \sqrt{17}}{17}$ B. $\frac{8}{9}$ C. $\frac{\sqrt{17}}{9}$ D. $\frac{\sqrt{17}}{8}$

Studdy Solution

STEP 1

What is this asking? If the sine of angle A is 8/9, and angles A and B are complementary in a right triangle, what's the cosine of angle B? Watch out! Don't mix up sine and cosine, and remember what "complementary" means!

STEP 2

1. Define Complementary Angles
2. Relate Sine and Cosine
3. Calculate Cosine B

STEP 3

Alright, so we're dealing with complementary angles.
What does that even mean?
It means they add up to $\mathbf{90^\circ}$ !
Since we're in a right triangle, and one angle is $\mathbf{90^\circ}$ , the other two must be complementary.
So, we know $\angle A + \angle B = 90^\circ$ .

STEP 4

STEP 5

Mathematically, we can write this as $\sin(A) = \cos(90^\circ - A)$ .
Since $\angle A + \angle B = 90^\circ$ , we can rewrite this as $90^\circ - A = B$ .
Therefore, $\sin(A) = \cos(B)$ .

STEP 6

We're given that $\sin(A) = \frac{8}{9}$ .
And we just figured out that $\sin(A) = \cos(B)$ .
So, guess what?
We're practically done!

STEP 7

If $\sin(A) = \frac{8}{9}$ and $\sin(A) = \cos(B)$ , then $\cos(B)$ must also be $\frac{8}{9}$ !

STEP 8

So, the cosine of angle B is $\frac{8}{9}$ , which corresponds to answer choice B!

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Studdy Solution

STEP 1

What is this asking? If the sine of angle A is 8/9, and angles A and B are complementary in a right triangle, what's the cosine of angle B? Watch out! Don't mix up sine and cosine, and remember what "complementary" means!

STEP 2

1. Define Complementary Angles2. Relate Sine and Cosine3. Calculate Cosine B

STEP 3

STEP 4

STEP 5

STEP 6

We're given that sin⁡(A)=89\sin(A) = \frac{8}{9}sin(A)=98​.And we just figured out that sin⁡(A)=cos⁡(B)\sin(A) = \cos(B)sin(A)=cos(B).So, guess what?We're practically done!

STEP 7

If sin⁡(A)=89\sin(A) = \frac{8}{9}sin(A)=98​ and sin⁡(A)=cos⁡(B)\sin(A) = \cos(B)sin(A)=cos(B), then cos⁡(B)\cos(B)cos(B) *must* also be 89\frac{8}{9}98​!

STEP 8

So, the cosine of angle B is 89\frac{8}{9}98​, which corresponds to answer choice B!

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Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. Define Complementary Angles
2. Relate Sine and Cosine
3. Calculate Cosine B

We're given that $\sin(A) = \frac{8}{9}$ .
And we just figured out that $\sin(A) = \cos(B)$ .
So, guess what?
We're practically done!

If $\sin(A) = \frac{8}{9}$ and $\sin(A) = \cos(B)$ , then $\cos(B)$ must also be $\frac{8}{9}$ !

So, the cosine of angle B is $\frac{8}{9}$ , which corresponds to answer choice B!